Are scientists normal? In today’s Secret Science Society adventure, Peter and Mary discover what the word means to scientists and why math can be important!
It was another typical Monday afternoon. Peter and Mary had finished lunch and were heading toward their favorite class. They didn’t like science class just because they wanted to be scientists; they also liked the way that their teacher, Mr. Medes, always managed to give them an interesting experiment to do. As a result, they frequently tried to predict what the experiment would be while they ate lunch. Today’s discussion had spilled over and continued as they walked to class.
“Well, it is Monday, so we know that it will be about math,” Mary said.
“Sure, but that doesn’t narrow it down much. Remember last week, when he had us coloring maps all class period?” Peter replied.
“And the week before, when we played with clay,” Mary said as they came to the classroom door. “Well, there’s only one way to find out – let’s go ask Mr. Medes!”
Mr. Medes was waiting for the class as they filed in and found their seats. As soon as all thirty-two students had come in, he started in on his lesson for the day.
“Welcome, welcome! Today is Monday, and that means math! For today’s lesson, we’re going to follow in the footsteps of that great gambler, Pascal. We are going to flip coins!”
“What does gambling have to do with math?” Peter asked.
“Believe it or not, most of the early work in statistics was done by Pascal and other gamblers as they sought a way to make more money. Pascal even invented a gambling machine that we know as the roulette wheel; he did it to see if he could create a perpetual motion machine. Instead, it turned into a perpetual money machine. But we won’t need anything that fancy today. We just need pennies.” As he said that, Mr. Medes passed out a penny to each of the students.
“Now, this is a very simple experiment. What we are going to do is flip the coins. If they are ‘fair coins’, which means if they are evenly balanced, then about half of them should come up heads and half should come up tails. Ready? Everybody flip!”
Suddenly the air was full of flipping coins. When the coins had stopped flipping, Mr. Medes asked for a show of hands from those who had heads.
“OK, we had 17 heads and 15 tails,” he said. “Let’s try that again, just to make sure that wasn’t a fluke.”
They repeated the test four more times, getting about the same distribution of heads and tails each time. Mr. Medes wrote 32 on the board then wrote the last result (14 heads and 18 tails) below it before turning back to the class.
“Good!” Mr. Medes said. “That proves that these are fair coins. We got about half heads and half tails each time, and everyone got a different set of heads and tails. But that raises an interesting question. What do you think would happen if we flipped the coins twice and kept track? How many times would we get either two heads or two tails in a row?”
“Well, you’ve got a fifty-fifty chance of getting heads after a tails, so you’d get it about half the time,” Mary said. “There are thirty two of us, so there’d be sixteen people with two heads or two tails.”
“No, you’re wrong,” Peter said. “You’ve only got a fifty-fifty chance of getting heads or tails on the first flip, so there should only be half as many. We’d have eight people with two heads or two tails. Everyone else will have a head and a tail or a tail and a head.”
“Well, there’s only one way to solve this problem,” Mr. Medes said. “Everyone take out a piece of paper and pencil to record your observations. Flip the coin and write down if it was heads or tails, then do it again.”
What do you think will happen? Do the experiment!
For a moment, the classroom was silent as everyone flipped their coins and wrote down the results. As soon as the last penny was put down, Mr. Medes spoke up again.
“OK, how many of you got two heads?” Nine hands went up, and Mr. Medes wrote that on the board. “How many got two tails?” Seven hands went up; Mr. Medes once again recorded the number, then put ‘16’ between them saying “And that leaves 16 people with either a head and a tail or a tail and a head.”
“Where did I go wrong?” Peter asked. “There should have been half that many!”
“Actually, you fell into the same trap that frustrated many of the mathematicians who discovered these rules. They forgot that everyone gets either a head or a tail for their first flip and that the real question is how many people will get a second head or tail. We can write it in math this way:”
Turning to the board, he drew
“Hey! That looks like the truth tables my mom uses when she programs!” Peter said.
“That’s right; this branch of math lies at the intersection of programming and statistics. It is also useful in genetics and many other fields. If you count it up, you’ll see that there are two boxes where you get a mixed result and two where you get a ‘pure’ result. So, if you had written it down like this, you’d have known the right answer without needing to do anything more. So – how can we decide how many ‘pure’ results we’ll get if we flip the coins three times?”
“We make another truth table!” Peter said.
“Right,” Mr. Medes replied as he drew a table on the board.
“We’ve got eight possible outcomes, and two of those are ‘pure’. So we should get one eighth of thirty two, or four, people with three heads in a row and the same number of people with three tails in a row; that means we should have eight ‘pure’ results. Now let’s flip the coins and see what happens. After you’ve flipped, raise your hand is you have three in a row of the same, heads or tails.”
Everyone again flipped their coins. The class looked around and burst into smiles as seven people raised their hands.
“Why didn’t we get exactly eight?” Mary asked, with a puzzled look.
“Because this is statistics, which works best for a large number of flips. If we had 320 people flipping the coins, we’d probably get somewhere between 75 and 85 pure results. If we had 3200 people flipping coins, we’d get somewhere between 780 and 820 pure results. As you flip the coin more often, random errors even themselves out and you come closer to the true value. Let’s try an experiment to see why. Mary, if you flip a coin, what are the odds of it coming up heads?”
“It will be heads half the time,” she replied.
“Right. Now Peter, flip your coin and tell me what it shows.”
Peter quickly flipped his coin. “It came up tails!”
“Now that doesn’t mean that Mary was wrong. What it means is that you don’t have enough flips, what a statistician calls ‘a significant population’, to decide what the odds are. The only way to tell the odds is to flip a coin a lot of times. And the more times you flip it, the bigger the population and the less chance that some error has crept into your results. As a matter of fact, a mathematician named John Kerrich tossed a coin 10,000 times while he was being held as a prisoner of war by the Nazis. He got heads 5,067 times and tails 4,933 times. If he had only flipped it once, he would have only gotten a head or a tail, just like Peter did. So bigger is better in statistics.”
At that point, the bell rang and Mr. Medes said “Don’t flip out – I’ll see you tomorrow for more science!”